Electron Gas Dynamic Conductivity Tensor on the Nanotube Surface in Magnetic Field

Abstract

Kubo formula was derived for the electron gas conductivity tensor on the nanotube surface in longitudinal magnetic field considering spatial and time dispersion. Components of the degenerate and nondegenerate electron gas conductivity tensor were calculated. The study has showed that under high electron density, the conductivity undergoes oscillations of de Haas-van Alphen and Aharonov-Bohm types with the density of electrons and magnetic field changes.

1. Introduction

Electromagnetic waves in the systems with cylindrical geometry have been studied for a long time [1–3]. The authors of the work [4] had solved the Maxwell’s equation taking into account the retardation for the conducting cylinder dipped in a dielectric medium. They considered time dispersion of the dielectric permittivity of conductor and dielectric. Also they had solved the dispersion equation for the wave spectrum and obtained new branches in the spectrum of surface polaritons. The paper [5] deals with polaritons in the magnetic wire with uniaxial anisotropy, dipped into a dielectric medium. Wire's dielectric and magnetic permittivity tensor frequency dependence also was studied. Polaritons spectra were found. Constant, parallel to the magnetic wire axis, external magnetic field effect was studied in paper [6]. In this paper, how wave characteristics allow one to obtain data about the material structure was shown.

A necessary of semiconductor nanotubes waveguide characteristics researching is caused by growing interest [7–10] to these systems. Plasma and magnetoplasma waves propagation on the nanotube surface were studied in [11–18]. They focused on cylindrical geometry systems calculations. These calculations were made by means of matter dielectric permittivity simple models. The problem is that authors of [11–18] have not used dielectric permittivity and conductivity tensors exact expressions. Also they have not taken into account the time and spatial dispersion. However, these exact formulas may significantly influence the tube waveguide characteristics expressions.

In this paper, we present conductivity tensor components calculation and their wave vector and frequency dependencies of the following system: electron gas on the nanotube surface, affected by a parallel to the tube axis external magnetic field. In Section 2, Kubo formula was obtained for the conductivity tensor. In Sections 3 and 4 we considered degenerate electron gas, and in Section 5, nondegenerate gas.

2. Dynamic Conductivity Tensor

Confined by the nanotube’s cylindrical surface, electron gas imbedded into a parallel to the cylinder axis magnetic field has axial symmetry. Therefore, it is suitable to characterize the electron state by the conserved quantities: angular momentum projection 𝑚=0,±1,…, impulse projection 𝑘 of electron on tube axis 𝑧, and spin quantum number 𝜎=±1. Electron stationary state |𝑚𝑘𝜎⟩ wave function and its energy spectrum are equal [11–14] toΨ_𝑚𝑘𝜎(𝜑,𝑧,𝛼)=1√𝑆𝑒^{𝑖(𝑚𝜑+𝑘𝑧)}𝜀𝜒_𝜎(𝛼),(1)_𝑚𝑘𝜎=𝜀₀𝑚+ΦΦ₀²+𝑘²2𝑚_∗+𝜎𝜇_𝐵𝐵,(2) where 𝑚_∗ and 𝜇_𝐵 are the effective mass and the electron spin magnetic moment, 𝜑 and 𝑧 are cylindrical coordinates, 𝜒_𝜎(𝛼) is a spin wave function, 𝜀₀=1/2𝑚_∗𝑎² is a rotational quantum, 𝑎 is the tube radius, Φ=𝜋𝑎²𝐵 is the magnetic induction 𝐵 flux through the tube section, Φ₀=2𝜋𝑐/𝑒 is flux quantum, 𝑆=2𝜋𝑎𝐿 is the nanotube lateral surface area, and the length of nanotube is 𝐿. Hereinafter, the quantum constant is considered to equal unity. The spectrum (2) is a set of one-dimensional subzones, whose boundaries 𝜀_𝑚0𝜎 are not equidistant.

One can expand the current electron density operator in field 𝐵 into series of cylindrical harmonics (1):𝐉(𝜑,𝑧)=_𝑚𝑘𝐉(𝑚,𝑘)Ψ_𝑚𝑘(𝜑,𝑧),(3) where 𝐉(𝑚,𝑘)=𝑒_{𝑚1𝑘1𝑚2𝑘2𝜎}⟨𝑚₁𝑘₁||𝐕(−𝑚,−𝑘)||𝑚₂𝑘₂⟩𝑎^+𝑚_1𝑘1𝜎𝑎_{𝑚2𝑘2𝜎}.(4) Here 𝑎_𝑚𝑘𝜎 and 𝑎^{+𝑚𝑘𝜎} are annihilation and creation operators of electrons in state |𝑚𝑘𝜎⟩,and ⟨𝑚₁𝑘₁|𝐕(−𝑚,−𝑘)|𝑚₂𝑘₂⟩ are operator 𝐕 matrix elements 𝐕(−𝑚,−𝑘)=12𝑚_∗−𝑖∇−𝑐𝑒𝐀Ψ^∗𝑚𝑘+Ψ_∗𝑚𝑘(𝜑,𝑧)(𝜑,𝑧)×−𝑖∇−𝑒𝑐𝐀(5) in the functional basis (1), 𝐁=rot𝐀. The operator’s (4) cylindrical components are equal to𝐽_𝜑(𝑚,𝑘)=𝑒𝑚_∗𝑎√𝑆_{𝑚𝑘𝜎}𝑚^+ΦΦ₀+𝑚2𝑎^{+𝑚𝑘𝜎}𝐽_𝑧𝑎_{(𝑚′+𝑚)(𝑘}^′_+𝑘)𝜎,(𝑚,𝑘)=𝑒𝑚_∗√𝑆_{𝑚𝑘𝜎}𝑘^+𝑘2𝑎^{+𝑚𝑘𝜎}𝑎_{(𝑚′+𝑚)(𝑘}^′_+𝑘)𝜎.(6)

Within the framework of electron gas linear response theory [19, 20] to the weak electric field 𝐄, changes with wave vector 𝑞 and frequency 𝜔, it is possible to obtain the current density𝑗_𝜇(𝑚,𝑞,𝜔)=_𝜈𝜎_𝜇𝜈(𝑚,𝑞,𝜔)𝐸_𝜈(𝑚,𝑞,𝜔),(7) where𝜎_𝜇𝜈(𝑚,𝑞,𝜔)=𝑖𝑒²𝑛𝑚_∗𝜔𝛿^𝜇𝜈+1𝜔^∞0𝑑𝑡𝑒^𝑖𝜔𝑡×𝐽_𝜇(𝑚,𝑞,𝑡),𝐽_𝜈(−𝑚,−𝑞,0),(8) Kubo formula for conductivity tensor. Here 𝐽_𝜇(𝑚,𝑞,𝑡) is a component of Heisenberg operator (6), 𝑛 is a surface electron density, [𝑎,𝑏]=𝑎𝑏−𝑏𝑎, and angle brackets mark Gibbs average.

We have substituted (6) in (8) and obtained the relation between the conductivity tensor and Fourier time component of two-electron Greens function [19, 20]:𝐾_𝑡(1,2;3,4)=−𝑖𝜃(𝑡)𝑎⁺¹(𝑡)𝑎₂(𝑡),𝑎⁺³(0)𝑎₄(0),(9) where 1=(𝑚₁,𝑘₁,𝜎₁),…, 𝜃(𝑡) is a Heaviside’s function. In case of free electron gas, function (9) is expressed by one-electron Green function. In the result, the conductivity tensor components (8) are equal to𝜎_𝜑𝜑(=𝑚,𝑖𝑞𝑒,₂𝜔)𝑛⋅𝑚_∗𝜔+𝑚^2∗𝑎𝑖𝑒²²𝜔𝑆_{𝑚′𝑘𝜎}𝑓𝜀_{𝑚′𝑘𝜎}𝑚^+Φ/Φ₀+𝑚/2²−𝜀𝑚_{𝑚′𝑘𝜎}−𝜀_{(𝑚′+𝑚)(𝑘+𝑞)𝜎}+𝜔+𝑖0_+Φ/Φ₀−𝑚/2²𝜎𝜀_{(𝑚′−𝑚)(𝑘−𝑞)𝜎}−𝜀_{𝑚′𝑘𝜎}+𝜔+𝑖0_𝜑𝑧,(=𝑚,𝜎𝑞,_𝜑𝑧𝜔)=𝑖𝑒(₂𝑚,𝑞,𝜔)𝑚^2∗𝑎𝜔𝑆_{𝑚′𝑘𝜎}⋅𝑚_𝑓𝜀_{𝑚′𝑘𝜎}+Φ/Φ₀+𝑚/2(𝑘+𝑞/2)−𝜀𝑚_{𝑚′𝑘𝜎}−𝜀_{(𝑚′+𝑚)(𝑘+𝑞)𝜎}+𝜔+𝑖0_+Φ/Φ₀−𝑚/2(𝑘−𝑞/2)𝜎𝜀_{(𝑚′−𝑚)(𝑘−𝑞)𝜎}−𝜀_{𝑚′𝑘𝜎}+𝜔+𝑖0_𝑧𝑧,(𝑚=,𝑖𝑞,𝑒𝜔₂)𝑛𝑚_∗𝜔+𝑚𝑖^2∗𝑒𝜔²𝑆_{𝑚′𝑘𝜎}⋅(𝑘+𝑞/2𝑓)₂𝜀_{𝑚′𝑘𝜎}𝜀−_{𝑚′𝑘𝜎}−𝜀(𝑘−𝑞/_{(𝑚′+𝑚)(𝑘+𝑞)𝜎}2)+₂𝜔+𝑖0𝜀_{(𝑚′−𝑚)(𝑘−𝑞)𝜎}−𝜀_{𝑚′𝑘𝜎}+𝜔+𝑖0,(10) where 𝑓(𝜀) is Fermi function. These components are included into the dispersion equation for electromagnetic waves, propagating along the tube.

In case of 𝑎→∞, one can obtain from (10) the well-known Drude-Lorentz expression, which describes two-dimensional electron gas with the magnetic field absence: 𝜎_𝜑𝜑=𝜎_𝑧𝑧=𝑖𝑒²𝑛/𝑚_∗𝜔, 𝜎_𝜑𝑧=0. The real part of conductivity tensor in this case is equal to zero. It is caused by not taking into account the electrons collisions. That effect can be considered by means of 𝜔→𝜔+𝑖𝜈_𝑐 replacement in (10). Here 𝜈_𝑐 is the collisions frequency.

3. Degenerate Gas

At zero temperature, integrals over 𝑘, included into (10), can be calculated exactly. Thus, real and imaginary parts of conductivity (10) are equal to Re𝜎_𝜑𝜑(𝑚,𝑞,𝜔)=−𝑒²×4𝜋_𝑚𝜎𝑚_∗||𝑞||𝑎³𝜔𝑚^+ΦΦ₀+𝑚2²×−𝜃𝑚𝐴₊₋₊−𝜃𝐴₋₋₊_+ΦΦ₀−𝑚2²Re𝜎_𝜑𝑧×𝜃𝐴₊₊₋−𝜃𝐴₋₊₋,(𝑚,𝑞,𝜔)=−𝑒²×4𝜋_𝑚𝜎(𝑞𝑎)²𝜔𝑚^+ΦΦ₀+𝑚×𝜃𝐴₊₋₊2𝜔−Ω⁺−𝑚′+Φ−𝜃𝐴₋₋₊Φ₀−𝑚×𝜃𝐴₊₊₋2𝜔−Ω⁻Re𝜎_𝑧𝑧−𝜃𝐴₋₊₋,(𝑚,𝑞,𝜔)=−𝑚^∗𝑒²×4𝜋||𝑞||³𝑎𝜔_𝑚𝜎−𝜔𝜔−−ΩΩ⁺₋²𝜃𝐴₊₋₊−𝜃𝐴₋₋₊Im𝜎_𝜑𝜑²𝜃𝐴₊₊₋−𝜃𝐴₋₊₋,(11)(𝑚,𝑞,𝜔)=𝑒²𝑛×𝑚_∗𝜔_𝑚𝜎−4𝜋²𝑚𝑒²_∗𝑞𝑎³𝜔𝑚′+ΦΦ₀+𝑚2²ln||||𝐴⁺⁻⁺𝐴₋₋₊−𝑚′+Φ||||Φ₀−𝑚2²ln||||𝐴⁺⁺⁻𝐴₋₊₋Im𝜎_𝜑𝑧||||,(𝑚,𝑞,𝜔)=−𝑒²𝑚𝑛×𝑚_{∗𝑚𝜎}𝑞𝑎𝜔−4𝜋²(𝑒𝑞²𝑎)²𝜔𝑚′+ΦΦ₀+𝑚×ln||||𝐴₊₋₊2𝜔−Ω⁺𝐴₋₋₊||||−𝑚′+ΦΦ₀−𝑚×𝜔−Ω₋2ln||||𝐴𝐴⁺⁺⁻₋₊₋Im𝜎_𝑧𝑧||||,(𝑚,𝑞,𝜔)=−2𝑒²𝜀₀𝑚²𝑛𝑞²𝜔−𝑚^∗𝑒²×4𝜋²𝑞³𝑎𝜔_𝑚𝜎𝜔−Ω⁺²ln||||𝐴𝐴⁺⁻⁺₋₋₊−𝜔−Ω₋||||²ln||||𝐴𝐴⁺⁺⁻₋₊₋||||,(12) where𝐴𝐴₊₋₊₊₊₋=𝑞𝜐^𝜎𝑚_′−𝜔₋+Ω₊,𝐴₋₋₊=−𝑞𝜐^𝜎𝑚_′−𝜔₋+Ω₊,=𝑞𝜐^𝜎𝑚_′−𝜐𝜔_+𝜎𝑚+Ω₋,𝐴₋₊₋=−𝑞𝜐^𝜎𝑚_′−𝜔₊+Ω₋,=2𝑚_∗𝜀_𝐹−𝜀_𝑚′𝜎^1/2(13) is the electron motion along the tube axis velocity maximum in subzone (𝑚′,𝜎), 𝜀_𝑚𝜎=𝜀_𝑚0𝜎 is the subzone boundary, 𝜀_𝐹 is the Fermi energy, 𝜔_±=𝜔±𝑞²/2𝑚_∗, andΩ_±(𝑚,𝑚′)=𝜀₀±𝑚′+ΦΦ₀±𝑚²∓𝑚′+ΦΦ₀²(14) is the frequency of the “vertical” electron transitions between the spectrum (2) subzones. While obtaining (12) out of (10), the identity below was used 𝐵²𝐵−𝐶=𝐵+𝐶+𝐵𝐶−²𝐶.(15)

4. Longitudinal Conductivity

This section concentrates on the analysis of degenerate electron gas longitudinal conductivity, (11) (12), assuming 𝑚=0. The numerical calculations were carried out also for 𝑚=1 case. In this case, electrons make only intra-subzone transitions without spin change. Longitudinal conductivity is included into the propagating along the nanotube axis intra-subzone magnetoplasma waves dispersion equation [13], because it is related to the polarization operator 𝑃 by the relation belowIm𝜎_𝑧𝑧(0,𝑞,𝜔)=𝑒²𝜔𝑞²Re𝑃(0,𝑞,𝜔).(16) In case of 𝑚=0, we obtain the following equations from (11) and (12): Re𝜎_𝑧𝑧(𝑞,𝜔)=−𝑒²𝑚_∗𝜔×4𝜋_𝑚𝜎||𝑞||³𝑎−𝜃𝜃𝑞𝑞𝜐𝜐^𝜎𝑚_𝜎𝑚−𝜔₋−𝜃−𝑞𝜐^𝜎𝑚−𝜔₋Im𝜎_𝑧𝑧−𝜔₊+𝜃−𝑞𝜐^𝜎𝑚−𝜔₊,(𝑞,𝜔)=−𝑒²𝑚_∗𝜔4𝜋²𝑞³𝑎_𝑚𝜎ln||||𝑞𝜐^𝜎𝑚−𝜔₋−𝑞𝜐^𝜎𝑚−𝜔₋−ln||||𝑞𝜐_𝜎𝑚||||−𝜔₊−𝑞𝜐^𝜎𝑚−𝜔₊||||.(17) Suppose that the flux ratio in (2) should be equal to Φ/Φ₀=𝑀+𝜂, where 𝑀=0,1,… is an integer part of Φ/Φ₀and 𝜂 is a fractional part (0≤𝜂<1). One can use the achievable in experiments values of carbon and semiconductor nanotube radii and obtain that the value of 𝑀 in the accessible magnetic fields is low, unless the tube radius is too large. Then the energy of electron (2) is minimal in the subzone (𝑚′=−𝑀, 𝜎=−1):𝜀_min=𝜀₀𝜂²−𝜇_𝐵𝐵.(18) In this case, the electron rotation energy is nearly compensated by the rotation in the magnetic field. Usually spin level splitting (2) is small, that is,2𝜇_𝐵𝐵<𝜀₀(1+2𝜂),(19) and the electron density corresponds to inequality𝑛<√𝑚_∗𝜇_𝐵𝐵𝜋²𝑎.(20) Then electrons partially fill only the lower subzone (−𝑀,−1). Therefore, (17) leads to the following:Re𝛿𝜎=−𝜋2𝑘𝑞⁰𝑥²Im𝛿𝜎=−𝑘₀,𝑥²2𝑞ln|||||1+𝑞/2𝑘⁰²−𝑥²1−𝑞/2𝑘₀²−𝑥²|||||,(21) where𝛿𝜎=𝜎_𝑧𝑧(𝑞,𝜔)𝑚^∗𝜔𝑒²𝑛,𝑥=𝑞𝜔𝜐₀𝜐₀,=𝑘⁰𝑚_∗=2𝑚_∗𝜀_𝐹+𝜇_𝐵𝐵−𝜀₀𝜂²^1/2(22) is the electrons velocity maximum in subzone (−𝑀,−1). Figures 1 and 2 show the 𝑥=𝜔/𝑞𝜐₀ dependencies of Re𝛿𝜎 and Im𝛿𝜎 under 𝜂=0.01,𝑞=𝑘₀,and 𝑘₀𝑎=1 in quantum limit, when Fermi energy is located in subzone (−𝑀,−1).

Figure 1 

Dependence of Re𝛿𝜎 on 𝑥=𝜔/𝑞𝜐0 under 𝜂=0.01, 𝑞=𝑘0, 𝑘0𝑎=1, 𝑚=0 (solid curve), and 𝑚=1 (dashed).

Figure 2 

Dependence of Im𝛿𝜎 on 𝑥=𝜔/𝑞𝜐0 under 𝜂=0.01, 𝑞=𝑘0, 𝑘0𝑎=1, 𝑚=0 (solid curve), and 𝑚=1 (dashed).

One can calculate what included into (17) sum by 𝑚′ using a Poisson formula, assuming 𝜀_𝐹≫𝜀₀, that is, a large quantity of subzones are filled. Within long wavelength assumption 𝑞𝜐_𝐹≪𝜔 (𝜐_𝐹 is a Fermi velocity), we obtain the following formula from (17): Im𝜎_𝑧𝑧=𝑒²𝑛𝑚_∗𝜔1+𝜋2²𝜀𝜀⁰_𝐹^∞3/4_𝑙=11𝑙^3/2×sin2𝜋𝑙𝜀_𝐹𝜀₀−𝜋4cos2𝜋𝑙ΦΦ₀.(23) Here we neglected the spin level splitting (2). Monotonous part of conductivity (23) under the given electron density does not depend on the tube radius and coincides with the two-dimensional electron gas conductivity [21]. It is within the 𝑎→∞, when the tube is cut along its generatrix and unrolled onto a plane. Conductivity (23) experiences the Aharonov-Bohm and de Haas-van Alphen oscillation types. The reason of the de Haas-van Alphen oscillation types existance is that the electron states density has root peculiarities on the subzone bounaries. Change of the tube radius or electron density causes these peculiarities to pass throw Fermi energy and de Haas-van Alphen oscillation types to appear. If we study conductivity dependence (23) on 𝜀^𝐹1/2=(𝑛/𝜈)^1/2 (𝜈 is the two-dimensional electron gas states density), then the oscillation period is 1/√2𝑚_∗𝑎. If we study the radius dependence, the period of conductivity oscillations is equal to 1/√2𝑚_∗𝜀_𝐹. Aharonov-Bohm oscillations are caused by the magnetic flux through the tube cross-section change. Their period is equal to the flux quantum. Figure 3 shows the dependence of the (23) first conductivity harmonic on (𝜀_𝐹/𝜀₀)^1/2 under Φ/Φ₀=0.01.

Figure 3 

Dependence of the first harmonic Im𝛿𝜎 (23) on (𝜀𝐹/𝜀0)1/2 under Φ/Φ0=0.01.

5. Nondegenerate Electron Gas

Nondegenerate electron gas emerges if inequality 𝛽|𝜇|≫1 is true on the semiconductor nanotube surface. Here 𝛽 is a reverse temperature and 𝜇 is a chemical potential. Using Boltzmann’s statistics, the chemical potential is related to the electron density as 𝑛=1𝜋^3/2𝑎𝑚_∗2𝛽𝑒^𝛽𝜇ch𝛽𝜇_𝐵𝐵^∞_𝑚_=−∞𝑒^{−𝛽𝜀𝑚},(24) where 𝜀_𝑚=𝜀₀(𝑚+Φ/Φ₀)². The included here summation may be transformed by means of formula [22] ^∞_𝑚=−∞𝑒^{−𝑥(𝑚+𝜐)2}=𝜋𝑥^∞_𝑙=−∞𝑒^{−𝜋2𝑙2/𝑥}cos(2𝜋𝑙𝜐),𝑥>0.(25) Real and imaginary tensor (10) parts in the studied case assuming 𝑚=0 are equal toR=e𝜎_𝜑𝜑(𝑞,𝜔)𝑒₂4√𝜋𝑚_∗𝛽𝜀₀^3/2||𝑞||𝑎³𝜔𝑒^𝛽𝜇×exp−𝑚_∗ch𝛽𝜇_𝐵𝐵𝛽2𝜔𝑞⁻²−exp−𝑚^∗𝛽×2𝜔𝑞⁺²_∞𝑙=_−∞exp−𝜋²𝑙²𝛽𝜀₀1−2𝜋²𝑙²Re𝜎_𝜑𝑧𝛽𝜀₀cos2𝜋𝑙ΦΦ₀,=(√𝑞,𝜔)𝜋𝑒²2𝛽𝜀₀^3/2(𝑞𝑎)²×_∞𝑒^𝛽𝜇ch𝛽𝜇_𝐵𝐵_𝑙=_−∞𝑙exp−𝜋²𝑙²×exp−𝑚_∗𝛽𝜀₀𝛽2𝜔𝑞⁻²−exp−𝑚^∗𝛽×sin2𝜋𝑙Φ2𝜔𝑞⁺²Re𝜎_𝑧𝑧Φ₀,=𝑒(𝑞,𝜔)₂𝑛𝜔||𝑞||³𝜋𝑚_∗𝛽×exp2−𝑚_∗𝛽2𝜔𝑞⁻²−exp−𝑚^∗𝛽Im𝜎_𝜑𝜑2𝜔𝑞⁺²,=𝑒(𝑞,𝜔)₂𝑛𝑚_∗𝜔+4𝜋𝑚_∗𝑒²𝛽𝜀₀^3/2𝑞𝑎³𝜔𝑒^𝛽𝜇×𝐹ch𝛽𝜇_𝐵𝐵𝑚_∗𝛽2𝜔𝑞⁻−𝐹𝑚_∗𝛽×2𝜔𝑞⁺_∞𝑙=_−∞exp−𝜋²𝑙²𝛽𝜀₀1−2𝜋²𝑙²Im𝜎_𝜑𝑧𝛽𝜀₀cos2𝜋𝑙ΦΦ₀,=(𝑞,𝑒𝜔)₂2𝛽𝜀₀^3/2(𝑎𝑞)²×_∞𝑒^𝛽𝜇ch𝛽𝜇_𝐵𝐵_𝑙=_−∞𝑙exp−𝜋²𝑙²×𝐹𝛽𝜀₀𝑚_∗𝛽2𝜔𝑞⁻−𝐹𝑚_∗𝛽×sin2𝜋𝑙Φ2𝜔𝑞⁺Im𝜎_𝑧𝑧Φ₀,=𝑒(𝑞,𝜔)₂𝑛𝜔𝑞³𝑚_∗𝛽×𝐹2𝑚_∗𝛽2𝜔𝑞⁻−𝐹𝑚_∗𝛽2𝜔𝑞⁺,(26) where𝐹(𝑥)=1√𝜋𝑝^∞_−∞𝑑𝑦𝑒^−𝑦2𝑥−𝑦.(27) Components 𝜎_𝜑𝜑 and 𝜎_𝜑𝑧 in (26) are expressed through 𝛽 and 𝜇, that is, they refer to an open system of tube surface electrons. They experience only Aharonov-Bohm oscillations with the magnetic field changes. Aharonov-Bohm oscillations are not taking place in 𝜎_𝑧𝑧 component while it is described in 𝛽 and 𝑛 terms.

6. Conclusion

Simplified conductivity models are usually used for studying electromagnetic waves propagation in the cylindrical geometry systems, for example, in nanotubes. The metal cylinder conductivity is often believed to be endless, and the dielectric permittivity of the matter, where cylinder is dipped, is considered to be constant or only frequency dependent. Conductivity’s spatial dispersion as a rule is not taken into account. Nonetheless, the electromagnetic field’s nature in the tube and its waveguide characteristics are sensitive to the surface currents. Therefore, the electron gas conductivity tensor components calculation problem, taking into account spatial and time dispersion, is worth consideration. It is given in this work conductivity tensor components may be used in obtaining a dispersion equation for electromagnetic waves in the tube. The example of longitudinal conductivity showed what data of electron gas is possible to obtain by measuring conductivity. Particularly, imaginary part of conductivity experiences de Haas-van Alphen and Aharonov-Bohm oscillations types with the electron density and magnetic field changes. The oscillation periods measurement enables one to identify the electron effective mass and the combination of universal constants included into flux quantum.

Acknowledgment

The authors would like to thank T. Rashba for help with preparation of the paper.